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2008•Journal of Shandong UniversityRequires access

Bounds for spectral radii of iterative matrices

Jing Liu

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Abstract

For solving a system of linear equations with the form of Ax=b,it often splits A into A=M-N,Where M is nonsingular.It is known that x(k+1)=M-1Nx(k)+M-1b(k=0,1,2,…) converges to the solution x=A-1b for each x(0) if and only if spectral radius ρ(M-1N)1. The matrix M-1N is called an iterative matrix.It is easy to see the estimates for bounds of ρ(M-1N) are interested.A concept of G-functions introduced by Hoffman and Nowosad was introduced to generalized G-functions,and the concept generalized Gfunctions were applied to obtain the bounds for moduli eigenvalues of the iterative matrices.The bounds for spectral radii of iterative matrices M1N were obtained and applied for solving linear systems.The obtained results improve the known corresponding results.Finally,a numerical example was given for illustrating the advantage of the results.

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For solving a system of linear equations with the form of Ax=b,it often splits A into A=M-N,Where M is nonsingular.It is known that x(k+1)=M-1Nx(k)+M-1b(k=0,1,2,…) converges to the solution x=A-1b for each x(0) if and only if spectral radius ρ(M-1N)1. The matrix M-1N is called an iterative matrix.It is easy to see the estimates for bounds of ρ(M-1N) are interested.A concept of G-functions introduced by Hoffman and Nowosad was introduced to generalized G-functions,and the concept generalized Gfunctions were applied to obtain the bounds for moduli eigenvalues of the iterative matrices.The bounds for spectral radii of iterative matrices M1N were obtained and applied for solving linear systems.The obtained results improve the known corresponding results.Finally,a numerical example was given for illustrating the advantage of the results.

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Available abstract

For solving a system of linear equations with the form of Ax=b,it often splits A into A=M-N,Where M is nonsingular.It is known that x(k+1)=M-1Nx(k)+M-1b(k=0,1,2,…) converges to the solution x=A-1b for each x(0) if and only if spectral radius ρ(M-1N)1. The matrix M-1N is called an iterative matrix.It is easy to see the estimates for bounds of ρ(M-1N) are interested.A concept of G-functions introduced by Hoffman and Nowosad was introduced to generalized G-functions,and the concept generalized Gfunctions were applied to obtain the bounds for moduli eigenvalues of the iterative matrices.The bounds for spectral radii of iterative matrices M1N were obtained and applied for solving linear systems.The obtained results improve the known corresponding results.Finally,a numerical example was given for illustrating the advantage of the results.

Key concepts: Spectral radius, Invertible matrix, Mathematics, Eigenvalues and eigenvectors, Iterative method, Matrix (chemical analysis), Applied mathematics, Linear system

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